bad reduction for elliptic curves

I would like to stress something that, I think, has not been point out very explicitly in any of the nice answers above.

The question that you are asking, if I understand it correctly, is a local question. So let's start with, say, a $p$-adic field $K$, and with the equation $f(x,y)=0$ of (the affine piece of) an elliptic curve $E$ with coefficients in the ring of integers $R$ of $K$.

To define the reduction of $E$ modulo the maximal ideal $mathfrak{m}$ of $R$, one cannot follow the naive approach of simply considering the reduction mod $mathfrak{m}$ of the equation $f(x,y)=0$.

Instead one should look at what Tate called a minimal Weierstrass equation $f_{rm min}(x,y)=0$ for $E$ (LNM 476 pag. 39), and define the reduction $bar E$ of $E$ as the object obtained by considering the reduction mod $mathfrak{m}$ of $f_{rm min}(x,y)=0$.

The point now is that the minimal equation is NOT stable by taking field extensions: if $L/K$ is a finite extension then $f_{rm min}(x,y)$ need not still be a minimal equation for the base change of $E$ to $L$ (but it is if $L/K$ is unramified). Therefore it might very well happens that while $f_{rm K, min}(x,y)=0$ has a singular reduction mod $mathfrak{m}$ the corresponding thing does not hold for $f_{rm L, min}(x,y)=0$ (the meaning of the subscripts is obvious I hope).

I tried to come up with an explicit example illustrating this phenomena. I failed to find one in a reasonable amount of time. But I am sure they can be found (easily?) in the literature.